The uniqueness problem of meromorphic mappings with deficiencies (Q1961354)

A classical theorem of Nevanlinna states that two meromorphic functions \(f\) , \(g\) on \(\mathbb{C}\) must coincide if there are five points \(p_1,\ldots, p_5\) on \({\mathbb P}_1\) such that \(f^{-1}(p_i)=g^{-1}(p_i)\) for all \(i=1,\ldots,5\). This result has been generalized to higher dimensions [see e.g. \textit{S. J. Drouilhet}, Trans. Am. Math. Soc. 265, 349-358 (1981; Zbl 0476.32034)]. The article under review is concerned with generalizations to higher dimensions where one prescribes not only the values at certain places but also deficiency (in the sense of value distribution theory). Let \(M\) be a projective manifold for which there exists a dominant meromorphic map \(f_0:{\mathbb C}^m\to M\). Let \(L\) be a big line bundle on \(M\), \(D\in|L|\) with only simple normal crossings. In this setup the author considers meromorphic mappings \(f:{\mathbb C}^m\to M\) with prescribed pre-images of \(D\) (up to a certain multiplicity), which coincide with \(f_0\) on these pre-images and furthermore have non-zero deficiency for at least one irreducible component of \(D\). He proves that there is only one such mapping under the assumption that \(L\) is sufficiently big and that \(D\) fulfills a certain genericity condition.